Which elements x^r in cyclic group Z_k have order k? (This will be a condition involving the two positive integers r and k.)

Example given by professor:

Consider the cyclic group of order k given by
Z_k = {1, x, x^2, x^3, ... , x^(k-1)},
where x^k = 1 = x^0 and where one defines

(x^i) * (x^j) := x^t, where i+j = m*k + t

for t in {0, 1, 2, ... , k-1} and m is an
integer not smaller than 0 ("nonnegative").
That is, "i+j is equal to t modulo k",
t is the remainder when one divides i+j by k.

The order of the element x is k since k is the
least positive integer such that x raised to the
power is 1. Order of the identity element isn't
defined as every power is the identity.

E.g., if k = 3, then (x^2) * (x^2) = x^1 = x,
while (x^1) * (x^2) = x^3 = x^0 = 1. Hence, the
powers of x^2 are the same as the powers of x
but in a different order. Indeed,
(x^2)^2 = (x^2) * (x^2) = x and
(x^2)^3 = (x^2)^2 * (x^2) = x * (x^2) = x^3 = 1
so the order of x^2 is equal to 3 which is the
same as the order of x.

2 Answers
2

Hint: in a cyclic group like $\langle\mathbb{Z}_k, + \rangle$ of order $k$, if $\gcd(k, r) = 1$, then $|x^r| = k$, $1\leq r \leq k$. Written additively, this means that in this specific group, all elements $rx$ (e.g., here $x = 1$) will have order $k$ if and only if $\gcd(r, k) = 1$. If and only if $r$ and $k$ are relatively prime.